be the altitude of a prism, which, having ABC for its base, is equal to their difference.

Divide the altitude AT into equal parts Ax, xy, yz, &c., each less than Aa, and let k be one of those parts; through the points of division, pass planes parallel to the plane of the bases: the corresponding sections formed by these planes in the two pyramids will be respectively equivalent, (B. VII, Prop. xiv,) namely, DEF to def, GHI to ghi, &c.

This being granted, upon the triangles ABC, DEF, GHI, &c., taken as bases, construct exterior prisms, having for edges the parts AD, DG, GK, &c., of the edge SA. In like manner, on the bases def, ghi, klm, &c., in the second pyramid, construct interior prisms, having for edges the corresponding parts of Sa. It is plain that the sum of all the exterior prisms of the pyramid SABC will be greater than this pyramid; and, also, that the sum of all the interior prisms of the pyramid Sabc will be less than this. Hence the difference between the sum of all the exterior prisms and the sum of all the interior ones, must be greater than the difference between the two pyramids themselves.

Now, beginning with the bases ABC, abc, the second exterior prism DEFG is equivalent to the first interior prism defa, because they have the same altitude k, and their bases DEF, def are equivalent: for like reasons, the third exterior prism GHIK, and the second interior prism ghid, are equivalent; the fourth exterior, and the third interior; and so on, to the last in each series. Hence all the exterior prisms of the pyramid SABC, excepting the first prism DABC, have equivalent corresponding ones in the interior prisms of the pyramid

Sabc: hence the prism DABC is the difference between the sum of all the exterior prisms of the pyramid SABC, and the sum of all the interior prisms of the pyramid Sabc. But the difference between these two sets of prisms has already been proved to be greater than that of the two pyramids, which latter difference we supposed to be equal to the prism aABC: hence the prism DABC must be greater than the prism aABC; but in reality it is less, for they have the same base ABC, and the altitude Ax of the first is less than Aa the altitude of the second. Hence the supposed inequality between the two pyramids cannot exist: hence the two pyramids SABC, Sabc, having equal altitudes and equivalent bases, are themselves equivalent.

PROPOSITION XVI.

THEOREM. Every triangular pyramid is the third of the triangular prism having the same base and altitude.

Let FABC be a triangular pyramid, ABCDEF a triangular prism of the same base and altitude: the pyramid will be equal to one-third of the prism.

Conceive the pyramid FABC to be cut off from the prism by a section made along the plane FAC, and there

E

B

will remain the solid FACDE, which may be considered as a quadrangular pyramid whose vertex is F, and base

the parallelogram ACDE. Draw the diagonal AD, and extend the plane FAD, which will cut the quadrangular pyramid into two triangular ones FACD, FADE. These two triangular pyramids have for their common altitude the perpendicular drawn from F to the plane ACDE; they have equal bases, the triangles ACD, ADE being halves of the same parallelogram: hence the two pyramids FACD, FADE are equal. But the pyramid FADE and the pyramid FABC have equal bases, ABC, DEF; they have also the same altitude, namely, the distance of the parallel planes ABC, DEF: hence the two pyramids are equal. Now the pyramid FADE has already been proved equal to FACD; hence the three pyramids FABC, FADE, FACD, which compose the prism ABCD, are all equal. Hence the pyramid FABC is the third part of the prism ABCD, which has the same base and the same altitude.

Cor. The solidity of a triangular pyramid is equal to a third part of the product of its base by its altitude.

PROPOSITION XVII.

THEOREM. Any pyramid is measured by a third part of the product of its base by its altitude.

Let S-ABCDE be a pyramid, having the altitude SO; then will it be measured by the base ABCDE into one-third of the altitude SO.

For, extending the planes SEB, SEC through the diagonals EB, EC, the polygonal pyramid SABCDE will be divided into several triangular pyramids, all

having the same altitude SO. But (B. VII, Prop. xvi,) each of these pyramids is measured by multiplying its base ABE, BCE or CDE by the third part of its altitude SO: hence the sum of these triangular pyramids, or the polygonal pyramid SABCDE, will be measured by the sum of the triangles ABE, BCE, CDE, or the polygon ABCDE, multiplied by SO. Hence every

A

E

S

B

pyramid is measured by a third part of the product of its base by its altitude.

Cor. 1. Every pyramid is the third part of the prism which has the same base and the same altitude.

Cor. 2. Two prisms having the same altitude, are to each other as their bases.

Scholium. The solidity of any polyedral body may be computed, by dividing the body into pyramids; and this division may be accomplished in various ways. One of the simplest is to make all the planes of division pass through the vertex of one solid angle; in that case, there will be formed as many partial pyramids as the polyedron has faces, minus those faces which form the solid angle whence the planes of division proceed.

PROPOSITION XVIII.

THEOREM. Two similar pyramids are to each other as the cubes of their homologous sides.

a

E

S

For, two pyramids being similar, the smaller may be placed within the greater, so that the solid angle S shall be common to both. In that position the bases ABCDE, abcde will be parallel; because, since the homologous faces are similar, the angle Sab is equal to SAB, and Sbc to SBC: hence the plane ABC is parallel to the plane abc. This granted, let SO be the perpendicular drawn from the vertex S to the plane ABC, and o the point where this perpendicular meets the plane abc from what has already been shown, (B. VI, Prop. xv,) we shall have

A

B

SO So SA: Sa :: AB : ab; and,

consequently, SO: So :: AB : ab.

But the bases ABCDE, abcde being similar figures, we have ABCDE: abcde :: AB2 : ab2.

Multiply the corresponding terms of these two proportions; there results the proportion,

ABCDE × SO : abcde × ¦ So :: AB3 : ab3.

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Now ABCDEX SO is the solidity of the pyramid SABCDE, and abcdex So is that of the pyramid Sabcde, (B. VII, Prop. xvII :) hence two similar pyramids are to each other as the cubes of their homologous sides.